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Calculus

# Differentiability: Level 4 Challenges

$\lambda=\left( \frac{f(5102)-f(2015)}{f'(c)} \right) \left( \frac{f^2(2015)+f^2(5102)+f(2015)f(5102)}{f^2(c)} \right)$

Let $$f:[2015,5102] \rightarrow [0,\infty)$$be any continuous and differentiable function. Find the value of $$\lambda$$, such that there exists some $$c\in [2015,5102]$$ which satisfies the equation above.

$f(x) = \displaystyle \prod_{k=1}^{\infty} \left( \dfrac{1+ 2\cos\left( \dfrac{2x}{3^k} \right)}{3}\right)$

Let $$f(x)$$ denote an function as described above. The number of points where $$|xf(x)| + | |x-2|-1|$$ is non-differentiable in $$x \in (0,3\pi)$$ is $$k$$, find $$k^2$$.

$$(\alpha_1, \beta_1), \ (\alpha_2, \beta_2), \ \cdots \ , \ (\alpha_n, \beta_n)$$ are the points on the curve $S \ : \ y=\sin (x+y) \ \ \forall x \in [-4\pi, \ 4\pi]$
such that the tangents at these points to $$S$$ are perpendicular to the line $l \ : \ (\sqrt {2}-1)x+y=0.$

Find the value of $\left\lfloor \displaystyle\sum_{k=1}^{n} |\alpha_k|-|\beta_k|\right\rfloor.$

Details and Assumptions

• $$\lfloor{\cdots}\rfloor$$ denotes the floor function (greatest integer function).

• Even though in the problem, a plural form of points is given, there may only be one such point that exists.

What is the largest possible number of integers $$n$$ such that for some fixed polynomial $$f(x)$$ of degree $$3$$ with integer coefficients, $$f(n)=n^{1000}$$?

Let $$f(x)$$ be a non-constant thrice differentiable function defined on real numbers such that $$f(x)=f(6-x)$$ and $$f'(0)=0=f'(2)=f'(5)$$. Find the minimum number of values of $$p \in [0,6]$$ which satisfy the equation $(f''(p))^2+f'(p)f'''(p)=0$ Details and Assumptions:

• $$f'(p)=\left( \frac{df(x)}{dx} \right)_{x=p}$$

• $$f''(p)=\left( \frac{d^2f(x)}{dx^2} \right)_{x=p}$$

• $$f'''(p)=\left( \frac{d^3f(x)}{dx^3} \right)_{x=p}$$

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